Mathematics for the interested outsider

Extraordinary Naturality

First, notice that if we’re enriching over (in “ordinary” categories) then means “take a morphism from to and follow it with “. On the other hand, means “first do , then follow it with a morphism from to “. This recipe gives us back exactly the old naturality square, so -natural transformations are exactly the ordinary natural transformations we’re familiar with!

So let’s take this reformulation of the naturality condition and tweak it. Instead of considering a family of arrows (in ) , let’s move the variable over from the left to the right and consider a family . Here, is an object of , and is a bifunctor. Now we say that the are the components of an “extraordinary -natural transformation” if the following diagram commutes:

This looks bizarre at first, though clearly it’s related to our revision of the enriched naturality diagram. It turns out that we’ve seen this sort of naturality before, though. If we read the diagram in , consider a monoidal category with duals, and use the functor , then this is exactly the sort of naturality we find in the duality arrows !

Dually, we can define extraordinary -naturality for a family of morphisms . Write out this diagram, and show that the duality arrows provide an example.

As another exercise, take these extraordinary naturality diagrams and work out the interpretation in explicitly. That is, actually start with some morphism in the upper left-hand corner, and evaluate it all around. When we did this for our new -naturality diagram above we got our old naturality squares back. What do we get for extraordinary -naturality?

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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